Combinatorix

A tangible interface that supports collaborative learning of probabilities

Teaching abstract concepts is notoriously difficult, especially when we lack concrete metaphors that map to those abstractions. Combinatorix offers a novel approach that combines tangible objects with an interactive tabletop to help students explore, solve and understand probability problems. Students rearrange physical tokens to see the effects of various constraints on the problem space; a second screen displays the associated changes in an abstract representation, e.g., a probability tree. Using participatory design, college students in a combinatorics class helped iteratively refine the Combinatorix prototype, which was then tested successfully with five students. Combinatorix serves as an initial proof-of-concept that demonstrates how tangible tabletop interfaces that map tangible objects to abstract concepts can improve problem-solving skills.

Introduction

Many decisions benefit from understanding probability, e.g., when a patient must interpret the meaning of a medical test result or when a politician must weigh the costs and benefits of a particular policy. Unfortunately, Tversky and Kahneman demonstrated that everyone, even professional statisticians, suffer from systematic biases in their intuitive judgements of probability. Students make a variety of identifiable mistakes when solving probability problems and even graduate students who plan to teach mathematics retain strong misconceptions.

The challenge is how to help students develop an intuitive grasp of these abstract concepts. We are particularly interested in combinatorics, a branch of probability that deals with the enumeration, combination, and permutation of sets of elements and their mathematical relationships, because it results in a combinatorial explosion: even simple problems result in hundreds of possibilities that cannot be represented simply with physical objects, virtual or otherwise.

Design Challenge

The original motivation for this project stemmed from observations of students in a university-level course in combinatorics. Faced with only paper and pencil, many had difficulty developing intuitions about probabilities and suffered from the ‘stereotype threat’ that they are poor in math. We hoped that letting students manipulate concrete objects while simultaneously observing the corresponding changes in deep structure, e.g. a probability tree, would reinforce their intuitions about the underlying mathematical principles. Our goal was to create an engaging and playful environment that avoids excessive mathematical notations and encourages discussion.

Combinatorix

Hardware

Combinatorix (Fig. 4) supports several input techniques: a camera detects the location of fiducial markers and a wiimote provides the position of multiple infra-red pens. A projector displays additional information around the tangible objects. The interactive surface is 60 x 45 cm. and can accommodate up to four students at the same time.

The Combinatorix setup: The webcam detects location of fiducial markers; the wiimote detects position of infra-red pens

Software

The underlying application is written in Java and uses the Reactivision engine to detect fiducial makers [7]. Additional libraries, e.g., wrj4P50, communicate with the wiimote. The system is modular and can easily accommodate the creation of additional operators for constraining the sample space.

The current version displays two kinds of information: first, the tabletop interface shows a specific number of placeholders for objects. Letters can be placed on those spots to form a new combination. At the same time, the remaining number of letters for each step is displayed on top of each placeholder. A second screen displays a probability tree reflecting the current state of the problem. Letters can easily be replaced by other elements, including virtual, laser-cut and 3D-printed physical objects. Combinatorix supports up to 10 tangible objects and 20 virtual ones.

User Study

We contrast two educational positions in our user study. The first one, a “tell-and-practice” approach, advocates direct instructions followed by practice exercises. The idea is to expose students to the “truth”, and then reinforce this first exposition with drilling exercises. The second approach (labeled “inventing”) suggests providing carefully designed activities to activate prior knowledge in students, which can be then confronted with experts’ explanation of a domain. The idea here is to have students formulate their own theory of a phenomenon, and then have them realize the many subtleties that differentiate their basic understanding of a concept with expert theories. The first approach is widely used in classrooms, while many researchers in the learning sciences advocate the second one.

Experimental design of the user study

We computed learning gains by subtracting students’ scores on the pre-test from their scores on the post-test. The scores in the post-test supports the main hypothesis: Students who completed a hands-on activity on an interactive tabletop and then watched a mini-lecture significantly outperformed students who first watched the lecture and then completed the hands-on activity: F(1,22) = 9.28, p < 0.01, Cohen’s d = 1.61 (mean for the “video-table” group = 2.23, SD = 1.77, the “table-video” group = 4.23, SD = 1.42). This effect remained significant when computed at the dyad level: F(1,10) = 7.73, p < 0.05.

Students’ scores on the pre-post tests

Conclusion

Our findings suggest that innovative technologies can have radically different effects on students’ learning depending on how they are integrated with traditional teaching practices. Choosing the wrong sequence of activities may impede students’ learning, whereas adopting a constructivist perspective is likely to foster knowledge building. In this study we were able to replicate previous results showing that TUIs increased students’ learning gains when used as a discovery-learning tool before traditional instruction (as opposed to a “tell-and-practice” type of instruction). Those results have implications for a wide range of educational approaches (e.g., classroom instruction, flipped classrooms, MOOCs): when correctly designed and implemented, TUIs can boost students’ learning by: preparing them for future learning, providing them with a fertile ground for socio-constructivist activities, supporting their exploration of a problem space, and increasing their engagement in hands-on tasks.

Acknowledgments

I would like to thank the Amir Lopatin Fellowship for funding this project; I would also thank Wendy MacKay and Paulo Blikstein for their support of this project.